What is the currently industry-standard algorithm used to generate large prime numbers to be used in RSA encryption?
I'm aware that I can find any number of articles on the Internet that explain how ...

Say I want a random 1024-bit prime $p$. The obviously-correct way to do this is select a random 1024-bit number and test its primality with the usual well-known tests.
But suppose instead that I do ...

In the article A Method for Obtaining Digital Signatures and Public-Key Cryptosystems, the original RSA article, it is mentioned that Miller has shown that n (the modulus) can be factored using any ...

As I understand it, the RSA algorithm is based on finding two large primes (p and q) and multiplying them. The security aspect is based on the fact that it's difficult to factor it back into p and q. ...

I would like to know if there is an algorithm to generate a RSA key at the state of the art of the present cryptanalysis.
Beside the key lenght I know there are some weakness in the choice of prime ...

In the 1978 RSA paper, it is recommended, among other things, to choose primes $p$ such that $(p-1)$ has a large prime factor $u$. This was motivated by Pollard's p-1 algorithm. Further, the authors ...

I take the definition of safe prime as: a prime $p$ is safe when $(p-1)/2$ is prime.
Safe primes of appropriate size are the standard choice for the modulus of cryptosystems related to the discrete ...

Although this has been extensively discussed around here, I'm curious whether my approach makes sense, or I should just stick to "the standard version".
I'm implementing some homomorphic encryption ...

I am looking into implementing Pohlig-Hellman exponentation cipher and I would like to know how secure that algorithm is? I am guessing it's security relates greatly to the prime number used in it. ...